GHP Formalism
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The GHP formalism (or Geroch–Held–Penrose formalism), also known as the compacted spin-coefficient formalism, is a technique used in the
mathematics of general relativity Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include numbe ...
that involves singling out a pair of null directions at each point of
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. It is a rewriting of the
Newman–Penrose formalism The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Sp ...
which respects the covariance of Lorentz transformations preserving two null directions. This is desirable for
Petrov Type In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold. It is mos ...
D spacetimes, including black holes in general relativity, where there is a preferred pair of degenerate principal null directions but no natural additional structure to fully fix a preferred Newman–Penrose (NP) frame.


Covariance

The GHP formalism notices that given a spin-frame (o^A,\iota^A) with o_A \iota^A = 1, the complex rescaling (o^A,\iota^A)\rightarrow (\lambda o^A, \lambda^ \iota^A ) does not change normalization. The magnitude of this transformation is a boost, and the phase tells one how much to rotate. A quantity of weight (p,q) is one that transforms like \eta \rightarrow \lambda^p \bar^q \eta. One then defines derivative operators which take tensors under these transformations to tensors. This simplifies many NP equations, and allows one to define scalars on 2-surfaces in a natural way.


See also

*
General relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
* NP formalism


References

* * Mathematical methods in general relativity {{math-physics-stub